As a seminarian, I served as a Master of Ceremonies with one of our auxiliary bishops, Bishop Frank Kane. This was a great opportunity as a seminarian to meet priests and see parishes around the Archdiocese during my time in formation. Another positive effect this had was the hours of time given to me to reflect on the Holy Spirit, since 98% of the Masses I served with the Bishop were Confirmation Masses.
No one will deny that out of the Blessed Trinity, the Holy Spirit gets the least amount of screen time. It's rare (except today on Pentecost) that priests actually preach on the Holy Spirit in theological language; parishes or churches named after the Holy Spirit are either Protestant or the left side version of SSPX; music written on the Holy Spirit are reserved for Pentecost or a Confirmation; and the people who pray to the Holy Spirit are often seen as "crazy" or "charismatic" and not taken seriously.
I try my best to preach on the Holy Spirit and I consider myself a conservative charismatic; however, there is only so much I can do since the Church doesn't say much about the Holy Spirit. I'm not a fan of making assumptions or starting theological debates during my homilies so I'm always left with preaching what the Church teaches...and she doesn't teach much on the Holy Spirit. The official study of the Holy Spirit is called Pneumatology. Pneuma is Greek for "breath" or "wind," so its the study of the Spirit. We read in Genesis 1:1 that a mighty "wind" swept over the waters before creation...this would be the Holy Spirit.
I think, for now anyway, the most important characteristic of the Holy Spirit is that it's the creating spirit. Obviously, God the father is the creator but it's the Holy Spirit who descends down for the work of creating. It's the Holy Spirit that binds together and gushes forth the Love between the Father and the Son. The Holy Spirit is the same Love (since God is Love) that binds together the two flesh into one in the Sacrament of Holy Matrimony and consecrates men ordained to the Sacrament of Holy Orders. The Spirit creates out of its own Love, who is God.
So when we pray to the Holy Spirit, we should bring to Him our prayers for creating something. If we are trying to start a new ministry in our parish or diocese, if we are trying to start a relationship, getting a new job, or discovering a cure for cancer we should offer those prayers up to the Holy Spirit.
It always helps to look at the model and most perfect example of what it means to be a disciple of Jesus, our Blessed Mother. We see in the Annunciation that Mary, while she did not know it, had a unique relationship with each person of the Blessed Trinity. Mary was praying to God the Father seeking His will, she received the Holy Spirit who conceived the Son within her, and she obviously received the Son as her child. We too need to have a relationship with each person of the Trinity. This takes time and work but in the end it will make us saints and that's the goal: holiness.